Problem 62 - Algebra for College Students

Question

Solve each equation, and check the solution. \(\frac{8 x}{3}-\frac{x}{2}=-13\)

Step-by-Step Solution

Answer

The solution is x = -6.

Step 1: Find a common denominator

To combine the fractions \(\frac{8x}{3}\) and \(\frac{x}{2}\), find a common denominator. The smallest common multiple of 3 and 2 is 6.

Step 2: Express each term with the common denominator

Rewrite each term to have a denominator of 6: \(\frac{8x}{3} = \frac{16x}{6}\) and \(\frac{x}{2} = \frac{3x}{6}\).

Step 3: Combine the fractions

Rewrite the equation as \( \frac{16x}{6} - \frac{3x}{6} = -13 \) and combine the terms on the left side: \( \frac{13x}{6} = -13 \).

Step 4: Solve for x

To isolate x, multiply both sides by 6: \( 13x = -78 \). Then divide both sides by 13: \( x = -6 \).

Step 5: Check the solution

Substitute \( x = -6 \) back into the original equation to verify: \(\frac{8(-6)}{3} - \frac{(-6)}{2} = -13 \).Simplify: \(-16 + 3 = -13\).Since both sides are equal, \( x = -6 \) is correct.

Key Concepts

common denominators

When solving equations with fractions, finding a common denominator is crucial. It allows us to combine fractions easily.

To find the common denominator, identify the least common multiple (LCM) of the denominators. In our example, the denominators are 3 and 2.

The LCM of 3 and 2 is 6. This means we'd rewrite each fraction with 6 as the denominator.

So, \(\frac{8x}{3}= \frac{16x}{6} \) and \(\frac{x}{2} = \frac{3x}{6} \). From here, combining the fractions becomes straightforward.

fractions in algebra

Handling fractions in algebra can seem tricky, but with practice, it gets easier. Key steps include finding common denominators and combining like terms.

In the given equation, we started with \( \frac{8x}{3} - \frac{x}{2} = -13 \).After adjusting to the common denominator, the fractions were rewritten: \( \frac{16x}{6} - \frac{3x}{6} \).

Now, they shared the same denominator, so combining them was easy. The result was \( \frac{13x}{6} = -13 \).

From here, converting to a simpler form, we isolated \( x \) by multiplying each side by 6 and then solving for \( x \).

checking solutions in equations

After finding a solution for an equation, it's essential to check if it's correct. This verification ensures no errors were made during calculations.

For our equation, we found \( x = -6 \). We substituted \( x = -6 \) back into the original equation to check: \( \frac{8(-6)}{3} - \frac{-6}{2} = -13 \).
Simplifying, we had \( -16 + 3 = -13 \).

Both sides matched, confirming that our solution \( x = -6 \) was indeed correct. Always remember this step, as it helps catch possible mistakes and reinforces understanding.